Flagellated bacteria swim in circles near a rigid wall

The rotation of bacterial flagella driven by rotary motors enables the cell to swim through fluid. Bacteria run and reorient by changing the rotational direction of the motor for survival. Fluid environmental conditions also change the course of swimming; for example, cells near a solid boundary draw circular trajectories rather than straight runs. We present a bacterium model with a single flagellum that is attached to the cell body and investigate the effect of the solid wall on bacterial locomotion. The cell body of the bacterium is considered to be a rigid body and is linked via a rotary motor to the elastic flagellum which is modeled by the Kirchhoff rod theory. The hydrodynamic interaction of the cell near a solid boundary is described using the regularized Stokes formulation combined with the image system. We show that the trajectories of the bacteria near a solid boundary are influenced by the rotation rate of the motor, the shape of the cell body, helical geometry, and elastic properties of the flagellum.

Figure 1

A schematic view of a single-flagellated bacterium near a rigid planar wall located at $z=0$. The flagellum of the cell is represented by a space curve $\mathbf{X}$ and its associated orthonormal triad $\left\{{\mathbf{D}}^1,{\mathbf{D}}^2,{\mathbf{D}}^3\right\}$. The surface of the spheroidal cell body is represented by two Lagrangian descriptions: ${\mathbf{X}}_{\mathrm{b}}$ as a massless boundary and ${\mathbf{Y}}_{\mathrm{b}}$ as a massive boundary. The distance between the center of mass of the cell body and the wall is denoted by $h\left(t\right)$, and the inclination angle between the major axis of the cell body ($-{\mathbf{E}}_3$) from the $z=0$ plane is denoted by $\theta \left(t\right)$ at time $t$.

Figure 2

Schematic geometry with the wall at $z=0$. Velocity is evaluated at $\mathbf{x}$ with the point force and torque centered at ${\mathbf{X}}_k$. The image point ${\mathbf{X}}_k^{\mathrm{im}}$ is introduced by the relation ${\mathbf{X}}_k={\mathbf{X}}_k^{\mathrm{im}}+2h_k{\mathbf{e}}_3$.

Figure 3

Comparisons between the image system method (solid lines with circles) and the target point method when $dw=1.0\delta$ (solid), $1.5\delta$ (dashed), $2.0\delta$ (dash-dotted), and $2.5\delta$ (dotted) in terms of the forward swimming speed $V_f\left(t\right)$ (a), distance from the wall $h\left(t\right)$ (b), inclination angle $\theta \left(t\right)$ (c), and distance from the center of mass of the cell body in the image system method to those of the target point method (d).

Figure 4

Motion of a bacterium viewed from side (a) and viewed toward the wall (b). The cell, initially positioned in parallel to the wall ($z=0$) with height $h\left(0\right)=1.0\mu \mathrm{m}$, swims toward the wall and follows a circular trajectory.

Figure 5

Time evolution of height $h\left(t\right)$ (a), inclination angle $\theta \left(t\right)$ (b), forward swimming speed $V_f\left(t\right)$ (c), and radius $R\left(t\right)$ of a circular trajectory (d) for various initial heights $h\left(0\right)$.

Figure 6

Time evolution of the height $h\left(t\right)$ (a), the inclination angle $\theta \left(t\right)$ (b), the forward swimming speed $V_f\left(t\right)$ (c), and the radius $R\left(t\right)$ of the circular trajectory (d) for various motor frequencies $\omega$ from 300 to 700 Hz. There is a limiting stable swimming pattern in each case. As we increase the motor frequency, the limiting values of $h\left(t\right)$, $\theta \left(t\right)$, and $V_f\left(t\right)$ all increase. The limiting value of radius $R\left(t\right)$ is independent of the frequency $\omega$.

Figure 7

Limiting values of steady circular swimming motions resulted from changes in the bending modulus $a_1=a_2$ (left panels) and the twist modulus $a_3$ (right panels). For each change of the modulus, five types of limiting values are measured: the height $h^{*}$ (a), (f), the forward swimming speed $V_f^{*}$ (b), (g), the radius $R^{*}$ of the circular trajectory (c), (h), the helical pitch (d), (i), and the helical radius (e), (j) of the rotating flagellum together with those of the initial flagellum in the equilibrium state (dashed lines). Filled circles represent the case with the default values of the elastic moduli.

Figure 8

Limiting values of steady circular swimming motions resulted from changes in four geometrical parameters; the diameter of the cell body $2A$ (a), the helical radius $r_0$ (b), the helical pitch $p_0$ (c), and the number of helical turns $N_{\lambda }$ (d). For each change of the parameter, three types of limiting values are measured: the separation gap $h^{*}-A$ (upper panel), the forward swimming speed $V_f^{*}$ (middle panel), and the radius $R^{*}$ of the circular trajectory (lower panel). Simulation results with the default values are marked with filled circles.

Figure 9

Trajectories of bacteria with five different body ratios $A/B$. The minor axis $B$ of the spheroidal cell body is fixed as $B=0.5\mu \mathrm{m}$ and the major axis $A$ is varied from 0.5 to $0.9\mu \mathrm{m}$. The motor frequency is set at $\omega =500$ Hz, and all bacteria are initially placed at the same location indicated by the black dot (“Start”). The final position of cells is captured when $t=1.0$ s. The wall is located at $z=0$, and the trajectories are viewed toward the wall from above.

Figure 10

Limiting values of forward swimming speed $V_f^{*}$ (a), height $h^{*}$ (b), inclination angle ${\theta }^{*}$ (c), and radius $R^{*}$ of the circular trajectory (d) as functions of the cell body ratio $A/B$, while keeping $B=0.5\mu \mathrm{m}$.

Figure 11

Critical values of helical turns $N_{\lambda }$ as the aspect ratio $A/B$ of the cell body changes. The cell body ratio $A/B$ is varied from 1.2 to $1.9\mu \mathrm{m}$, while the minor axis $B$ of the spheroidal cell body is fixed as $B=0.5\mu \mathrm{m}$. The helical pitch and radius are fixed at $1.25\mu \mathrm{m}$ and $0.1989\mu \mathrm{m}$, respectively. The motor frequency is set at $\omega =500$ Hz, and the initial height is given at $h\left(0\right)=1.5\mu \mathrm{m}$. The circles in blue represent the case of entrapped cells, and the crosses in red represent the case of escaping cells. The solid line in black is an interpolated curve that separates the escape zone (shaded area) from the entrapment zone (white area).